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Exercises



1.
There are two basic forms of sleep: slow wave sleep (SWS) and rapid eye movement (REM) sleep. Infants spend about 50% of their sleep time in SWS and 50% in REM sleep. Adults below age 60 spend about 20% of their sleep time in REM and 80% in SWS sleep. In a study of sleep patterns, data was collected on 13 elderly males over age 60. The percentage of total sleep time spent in REM sleep is presented below.
21, 20, 22, 7, 9, 14, 23, 9, 10, 25, 15, 17, 11
(a)
Calculate the sample average and standard deviation.
(b)
The sample average is how far below 20%?
(c)
Calculate the standard error (SE) of the sample average.
(d)
The sample average is how many SE's below 20% ?
(e)
Is the sample average significantly below 20%, or is it just chance variation?
(f)
If you conduct a test of significance on the following hypothesis: `Does the data provide scientific evidence that elderly males spend less than 20% of their sleep time in REM?', how would you write the null and alternative hypotheses?
(g)
What is the test statistic you would use?
(h)
What distribution curve would you use to compute the P-value?
(i)
Calculate a (one-tailed) P-value for your test.
(j)
What is the conclusion of your test?

2.
Twenty-five factory workers were asked how many vacation days they take a year. The average of the sample was 22.85 days, and the standard deviation of the sample was 5.80.
(a)
Calculate the standard error of the sample average.
(b)
In the past, the company's Human Resources Office uses an average of 18 vacation days per worker in order to make available man-hour and payroll predictions. Do you think this standard of 18 days needs to be changed? (Is the the observed average of 22.85 significantly different from 18? Calculate a (one-tailed) P-value to measure statistical significance of the difference.)

3.
A new breakfast cereal Frosted Corn is test marketed for one month. The total sales for the first 9 quarters (3-month periods) indicate an average of $8350, with a standard deviation of $1840.
(a)
Treating the first 9 quarters as a sample of size 9, calculate a standard error for the average sales of $8350.
(b)
Based on a cost-profit analysis, production of Frosted Corn will be discontinued if the sample average of the first 9 quarters falls below $9000 with statistical significance. Does the data indicate production will be stopped? (Calculate a one-tailed P-value to measure statistical significance.)

4.
A Michigan automobile insurance company would like to estimate the average size of an accident claim for the fiscal year 2002. A random sample of 20 accident claims yielded an average of $1600 with a standard deviation of $800.
(a)
What is the standard error of the $1600 estimate?
(b)
The national average claim for the industry is $1500. Does the sample indicate that the Michigan average is different from the national average? (Conduct a two-tailed test of significance. What is the P-value of your test?)

5.
In order to test a new production method, 15 employees were selected randomly to try the new method. The mean production rate for the sample was 80 parts per hour (with sample SD 10 parts per hour).
(a)
Calculate a standard error for the sample mean production rate for the new method.
(b)
The mean production rate under the old method is known to be 70 parts per hour. Is the new method rate significantly different, or is this just luck-of-the-draw on the employees sampled ? (Calculate a two-tailed P-value for your test.)

6.
Safe Lumber Company hires an independent contractor to monitor the groundwater within a one mile radius of its factory to ensure that no toxic levels of arsenic are being released. The contractor takes a sample of groundwater from selected sites within the one mile radius. Safe Lumber will need to pay for an expensive clean-up if the average arsenic level in groundwater is shown to be over 50 parts per billion with statistical significance (using a one-tailed test).
(a)
Suppose that data from 9 sites had arsenic levels averaging 75 ppb with a sample SD of 30 ppb. Is this statistically significant proof that the average level is over 50 ppb?
(b)
If Safe Lumber wanted to avoid an expensive clean-up, should it prefer a one-tailed or two-tailed test? (What is the P-value if a two-tailed test were done?)
(c)
If Safe Lumber wanted to avoid an expensive clean-up, should it prefer a sample size of 9 or 25? (What is the P-value if the sample size was 25 instead of 9, with sample average and SD remaining the same?)
(d)
If Safe Lumber wanted to avoid an expensive clean-up, would it prefer to sample from high ground areas or low ground areas?
(e)
If Safe Lumber wanted to avoid an expensive clean-up, would it prefer to sample closer or farther from the factory?

7.
A personnel agency is comparing salaries of uncertified accountants versus CPA's. Among recent accountants that they placed, 4 were uncertified while 6 were CPA's. The annual salaries (in dollars) are summarized in the following table:

  Uncertified CPA
Average: $48,500 $56,600
SD: $8,700 $9,100
Sample size: 4 6

(a)
Are CPA salaries significantly higher than uncertified, or can this be just luck-of-the-draw? (Conduct a one-tailed test of significance using a pooled-SD t-test. What is the calculated P-value of the test?)
(b)
Suppose that the samples sizes were 40 and 60 instead of 4 and 6 (sample average and SD's the same), will your P-value in (a) get smaller or larger? What is the size of the new P-value?)
(c)
Suppose that you conduct a one-tailed unpooled-SD z-test for the sample sizes 4 and 6, do you expect the P-value to be smaller, larger, or approximately equal to the P-value in (a)? Which P-value is more correct ?
(d)
Suppose that you conduct an unpooled-SD z-test for the sample sizes 40 and 60, do you expect the P-value to be smaller, larger, or approximately equal to the P-value in (a)? Which P-value is more correct ?

8.
Do credit cards with no annual fee charge higher interest rates than cards that have annual fees? Among 29 cards surveyed, 17 had no annual fees while 12 charged an annual fee. Among the cards with no annual fee, the average interest rate was 19% (SD=8%). Among cards with an annual fee, the average interest rate was 17% (SD=3%).
(a)
Are interest rates significantly higher for cards with no annual fees? (Conduct a one-tailed test of significance using a pooled-SD t-test. What is the calculated P-value of the test?)
(b)
Conduct a one-tailed unpooled-SD z-test. Do you expect the P-value to be smaller, larger, or approximately equal to the P-value in (a)? Which P-value is more correct ?

9.
A new gasoline additive is supposed to make gas burn more cleanly and increase gas mileage in the process. Consumer Protection Anonymous conducted a mileage test to confirm this. They took seven of their cars, filled it with regular gas, and drove it on I-94 until it was empty. They repeated the process using the same cars, but using the gas additive. The recorded gas mileage follows:
Car 1 2 3 4 5 6 7
Without Additive 22 15 18 28 12 25 18
With Additive 26 19 17 34 17 25 22
(a)
For each of the 7 cars, calculate the mileage difference (With-Without), between the two fuel types.
(b)
Calculate the average and SD for the differences in (a).
(c)
Does the data support the claim of higher gas mileage? (Conduct a one-tailed test of significance. What is the P-value of your test?)
(d)
Suppose that the data were quadrupled, but the data just repeats itself four times (i.e. (22, 26),$\ldots$,(18, 22) each appears four times). Then the new sample size is 28. What is the new average and SD for the differences? What is the new P-value for the one-tailed test of significance??
(e)
Suppose that a t-test for independent samples were used on the original data with 7 cars. Do you expect the P-value to be smaller, larger, or approximately equal to the paired-t P-value? Which P-value is more correct ?
(f)
Eyeball the following idealized data and state any pattern that you see. (Without any statistical testing, what is the obvious conclusion about the effectivity of the gas additive?) Now compare the P-values for the paired-t test versus the t-test for independent samples. Which P-value is more correct ?
Car 1 2 3 4 5 6 7
Without Additive 22 15 18 28 12 25 18
With Additive 26 19 22 32 16 29 22

10.
Some stock market analysts have speculated that parts of West Michigan Telecom might be worth more than the whole. For example, the company's communication systems in Ann Arbor and Detroit could be sold to other communications companies. Suppose that a stock market analyst chose 9 acquisition experts and asked each to predict the return (in percent) on investment in the company held to the year 2003 if (i) it does business as usual, or (ii) if it breaks up its communication systems and sells all its parts. Their predictions follow:
Expert 1 2 3 4 5 6 7 8 9
Not Break Up 12 21 8 20 16 5 18 21 10
Break Up 15 25 12 17 17 10 21 28 15
(a)
On the average, do experts believe breaking up the company will earn a higher return? (Conduct a one-tailed test of significance. What is the P-value of your test?)
(b)
Suppose that a test for independent samples is used. Do you expect the P-value to be smaller, larger, or approximately equal to the P-value in (a)? Which P-value is more correct ?

11.
A survey conducted by USA Today showed that 118 of 250 investors owned some real estate.
(a)
What proportion of the surveyed investors owned some real estate?
(b)
Calculate a standard error (SE) for your estimate in (a).
(c)
Conduct a test of significance for H0: p=.5 versus $H_1: p\neq .5$. What is the P-value of your test?
(d)
Suppose a larger survey showed 1180 out of 2500 investors owned some real estate. What is the new sample proportion and SE? What is the new P-value for the test of significance above?

12.
A survey conducted by USA Today showed that 118 of 250 investors owned some real estate.
(a)
What percentage of the surveyed investors owned some real estate?
(b)
Calculate a standard error (SE) for your estimate in (a).
(c)
Conduct a test of significance for H0: p=.5 versus $H_1: p\neq .5$ using the sample percentage in the test statistic. What is the P-value of your test?

13.
An appliance manufacturer offers maintenance contracts on its major appliances. A manager wants to know what fraction of buyers of the company's convection ovens are also buying the maintenance contract with the oven. From a random sample of 120 sales slips, 31 of the oven buyers opted for the contract.
(a)
The proportion of customers who buy the contract along with their oven is estimated as _________.
(b)
Calculate a standard error for the estimate in (a).
(c)
Test the hypothesis H0: p=.3 versus the alternative $H_1: p\neq .3$. What is the P-value of your test?

14.
Researchers concerned if doctors were consistently adjusting dosages for weight of elderly patients studied 2000 prescriptions. They found that for 600 of the prescriptions, the doctors failed to adjust the dosages.

(a)
Doctors fail to adjust dosage for an estimated _________ percent of prescriptions.
(b)
Calculate a standard error for the percentage in (a).
(c)
Test the hypothesis $H_0: p\geq 1/3$ versus the alternative H1: p < 1/3. What is the P-value of your test?

15.
A believer in horoscopes is willing to put his beliefs to a test. You will present him each week with three horoscopes for that week, one of them being his own. At the end of each week he is to pick out the horoscope best describing his experiences.

(a)
Suppose he made 12 correct identifications out of 26 during the first 6 months. Should you concede that his faith has some justification, or can this be explained by guessing? (Use a one-tailed test.)
(b)
During the next six months, he again got 12 out of 26 correct (he now has 24 out of 52 total). Now, should you concede that his faith has some justification?

16.
According to a survey by Town Plaza Hotels, 12% of 150 men who were travelling on business brought a friend or spouse. In contrast, 28% of 101 women in the survey who were travelling on business brought a friend or spouse.
(a)
According to the data, there is an estimated difference of _________ between the the percentage of men and women who brought a friend or spouse along on business travel.
(b)
Calculate a standard error for the estimate in (a).
(c)
Conduct a test of significance for the difference between the two percentages. (Use a two-tailed test.)

17.
Borrowers Bank classifies credit cardholders as "prime" whenever they pay interest charges for at least 6 of the last 12 billing cycles. The table below cross classifies 2080 cardholders according to whether they are (i) prime or not, and (ii) below age 30 or not

  Below 30 30 and Over
Prime 475 510
Not Prime 376 719

(a)
The percentage of prime cardholders among those who are less than 30 years old is estimated as _________ % give or take _________ % or so.
(b)
The percentage of prime cardholders among those who are 30 years or older is estimated as _________ % give or take _________ % or so.
(c)
Are the two percentages in (a) and (b) significantly different?. (Use a two-tailed test.)


next up previous contents index
Next: Testing Equality of Frequencies Up: Testing Hypotheses Previous: Comparing two means (paired

2003-09-08